A geometrical property ofC(K) spaces

We introduce a geometrical property of norm one complemented subspaces ofC(K) spaces which is useful for computing lower bounds on the norms of projections onto subspaces ofC(K) spaces. Loosely speaking, in the dual of such a space ifx* is a w* limit of a net (x _a ^* ) andx*=x*₁+x*₂ with ‖x*‖=‖x*₁‖ + ‖x*₂‖, then we measure how efficiently thex _a ^* 's can be split into two nets converging tox*₁ andx*₂, respectively. As applications of this idea we prove that if for everyε>0,X is a norm (1+ε) complemented subspace of aC(K) space, then it is norm one complemented in someC(K) space, and we give a simpler proof that a slight modification of anl ₁-predual constructed by Benyamini and Lindenstrauss is not complemented in anyC(K) space. Research partially supported by a grant of the U.S.-Israel Binational Science Foundation. Research of the first-named author is supported in part by NSF grant DMS-8602395. Research of the second-named author was partially supported by the Fund for the Promotion of Research at the Technion, and by the Technion VPR-New York Metropolitan Research Fund.     

A logarithmic Gauss curvature flow and the Minkowski problem

Let X₀ be a smooth uniformly convex hypersurface and f a postive smooth function in Sⁿ. We study the motion of convex hypersurfaces X(·,t) with initial X(·,0)=θX₀ along its inner normal at a rate equal to log(K/f) where K is the Gauss curvature of X(·,t). We show that the hypersurfaces remain smooth and uniformly convex, and there exists θ^*>0 such that if θ<θ^*, they shrink to a point in finite time and, if θ>θ^*, they expand to an asymptotic sphere. Finally, when θ=θ^*, they converge to a convex hypersurface of which Gauss curvature is given explicitly by a function depending on f(x).     

w* sequential convergence

LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ _n ^* } _n=1 ^∞ ?X* which arew* converging to 0 while inf _n ‖x* _n ‖>0, are constructed.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

Green's and Dirichlet spaces associated with fine Markov processes

This is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space $\text{K}$ and the Dirichlet space $\text{H}$ associated with a symmetric Markov transition function p_t(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space $\text{H}$ can be built from functions which are right continuous along almost all paths; (b) the positive cone $\text{K}$⁺ in $\text{K}$ can be identified with a cone M of measures on the state space; (c) the positive cone $\text{H}$⁺ in $\text{H}$ can be interpreted as the cone of Green's potentials of measures μ?M. To every measurable set B in the state space E there correspond a subspace $\text{K}$(B) of $\text{K}$ and a subspace $\text{H}$(B) of $\text{H}$. The orthogonal projections of $\text{K}$ onto $\text{K}$ and of $\text{H}$ onto $\text{H}$(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ?M.     

The pexiderized Go?a?b-Schinzel functional equation

Let X be a linear space over a commutative field K. We characterize a general solution f,g,h,k:X→K of the pexiderized Go?a?b-Schinzel equation f(x+g(x)y)=h(x)k(y), as well as real continuous solutions of the equation.     

An existence-convergence theorem for a class of iterative methods

LetM ₀, characterized byx _k+1=G ₀(x _k),k?0,x ₀ prescribed, be an iterative method for the solution of the operator equationF(x)=0, whereF:X → X is a given operator andX is a Banach space. Let ω:X → X be a given operator, and let the methodM _mbe characterized byx _x+1,m =G _m(x _k,m),k?0,x _0,m prescribed, where $$G_i (x) = G_0 (x) - \sum\limits_{j = 0}^{i - 1} { F'(\omega (x))^{ - 1} F(G_j (x)), i = 1, . . . ,m,} $$ in whichG ₀:X → X is a given operator andF′:X → L(X) is the Fréchet derivative ofF. Sufficient conditions for the existence of a solutionx* _m ofF(x)=0 to which the sequence (x _k,m) generated from methodM _mconverges are given, together with a rate-of-convergence estimate.     

Characterization of the closed convex hull of the range of a vector-valued measure

For a set K in a locally convex topological vector space X there exists a set T, a σ-algebra $\text{S}$ of subsets of T and a σ-additive measure m: $\text{S}$ → X such that K is the closed convex hull of the range {m(E): E ∈ $\text{S}$} of the measure m if and only if there exists a conical measure u on X so that K K_u,K_u, the set of resultants of all conical measures v on X such that v < u.     

Classifying Hilbert bundles. II

A Hilbert bundle (p, B, X) is a type of fibre space p: B → X such that each fibre p^?1(x) is a Hilbert space. However, p^?1(x) may vary in dimension as x varies in X, even when X is connected. We give two “homotopy” type classification theorems for Hilbert bundles having primarily finite dimensional fibres. An (m, n)-bundle over the pair (X, A) is a Hilbert bundle over (p, B, X) such that the dimension of p^?1(x) is m for x in A and n otherwise. As a special case, we show that if X is a compact metric space, C₊X the upper cone of the suspension SX, then the isomorphism classes of (m, n)-bundles over (SX, C₊X) are in one-to-one correspondence with the members of [X, V_m($\text{C}$ⁿ)] where V_m($\text{C}$ⁿ) is the Stiefel manifold. The results are all applicable to the classification of separable, continuous trace C¹-algebras, with specific results given to illustrate.     

A convergence theorem for asymptotic contractions

We show that to each asymptotic contraction T with a bounded orbit in a complete metric space X, there corresponds a unique point x _* such that all the iterates of T converge to x _*, uniformly on any bounded subset of X. If, in addition, some power of T is continuous at x _*, then x _* is a fixed point of T. Dedicated to Professor Felix E. Browder with admiration and respect     

On implicit function theorems at abnormal points

We consider the equation F(x, σ) = 0, x ∈ K, in which σ is a parameter and x is an unknown variable taking values in a specified convex cone K lying in a Banach space X. This equation is investigated in a neighborhood of a given solution (x*, σ*), where Robinson’s constraint qualification may be violated. We introduce the 2-regularity condition, which is considerably weaker than Robinson’s constraint qualification; assuming that it is satisfied, we obtain an implicit function theorem for this equation. The theorem is a generalization of the known implicit function theorems even in the case when the cone K coincides with the whole space X.     

Existence and continuity of an implicit function in the neighborhood of an abnormal point

Solutions to the equation F(x, ??) = 0 with unknown x and the parameter ?? in the neighborhood of the solution (x _*, ??_*) under the additional constraint x ?? U, where U is a closed convex set, are studied. The sufficient conditions for existence of an implicit function without prior assumption of the normalcy of point x _* are given. The obtained result is used to investigate the local solvability of controlled systems with mixed constraints.     

Banach spaces in which a theorem of orlicz is not true

Let the Banach space X be such that for every numerical sequencet _n ↘0 there exists in X an unconditionally convergent series σx_n, the terms of which are subject to the condition ∥x_n∥=t_n (n=1,2,...). Then $$\mathop {sup}\limits_n \mathop {inf}\limits_{X_n } d(X_n ,l_\infty ^n )< \infty ,$$ where X_n ranges over all the n-dimensional subspaces of X.     

The Degree and Regularity of Vanishing Ideals of Algebraic Toric Sets Over Finite Fields

Let X* be a subset of an affine space 𝔸 ^s , over a finite field K, which is parameterized by the edges of a clutter. Let X and Y be the images of X* under the maps x → [x] and x → [(x, 1)], respectively, where [x] and [(x, 1)] are points in the projective spaces ? ^s?1 and ? ^s , respectively. For certain clutters and for connected graphs, we were able to relate the algebraic invariants and properties of the vanishing ideals I(X) and I(Y). In a number of interesting cases, we compute its degree and regularity. For Hamiltonian bipartite graphs, we show the Eisenbud–Goto regularity conjecture. We give optimal bounds for the regularity when the graph is bipartite. It is shown that X* is an affine torus if and only if I(Y) is a complete intersection. We present some applications to coding theory and show some bounds for the minimum distance of parameterized linear codes for connected bipartite graphs.     

Asymptotic centers and best compact approximation of operators intoC(K)

The existence of best compact approximations for all bounded linear operators fromX intoC(K) is related to the behavior of asymptotic centers inX ^*. IfK is just one convergent sequence, the condition is that everyω ^*-convergent sequence inX ^* will have an asymptotic center. We first study this property, solving some open problems in the theory of asymptotic centers. IfK is more “complex,” the asymptotic centers should behave “continuously.” We use this observation to construct operators fromC[0,1] intoC(ω ²) and from ?₁ intoL ₁ without best compact approximation. We also construct spacesX ₁,X ₂, isomorphic to a Hilbert space, and operatorsT ₁,∶X ₁→C(ω ²),T ₂∶?₁→X ₂ without best compact approximations.     

Construction and regularity of transition functions on polish spaces under measurability conditions

This paper concerns the construction and regularity of a transition (probability) function of a non-homogeneous continuous-time Markov process with given transition rates and a general state space. Motivating from a lot of restriction in applications of a transition function with continuous (in t≥0) and conservative transition rates q(t, x, Λ), we consider the case that q(t,x,Λ) are only required to satisfy a mild measurability (in t≥0) condition, which is a generalization of the continuity condition. Under the measurability condition we construct a transition function with the given transition rates, provide a necessary and sufficient condition for it to be regular, and further obtain some interesting additional results.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

Equivalent Gaussian measure whose R-N derivative is the exponential of a diagonal form

A simple necessary and sufficient condition, on a trace-class kernel K, is given in order to demonstrate the existence of a measurable (relative to the completed product σ-algebra) Gaussian process with covariance K. Using this result, sufficient conditions are given on the means and the covariances (relative to two equivalent (~) Gaussian measures P and P_λ) of a process X so that the Radon-Nikodým (R-N) derivative $\text{dp}{}_{\mathrm{\lambda }}\text{dP}$ is the exponential of the diagonal form in X. Analogues of the last two results in the setup of Hilbert space are also proved.     

Über Folgen,die BezÜglich Eines Mittels Rekurrent Sind

Let X be a convex subset of a finite-dimensional real vector space. A function M: X ^k → X is called a strict mean value, if M(x₁,…, x_k) lies in the convex hull of x₁,…, x_k), but does not coincide with one of its vertices. A sequence (x_n)_{n∈ ?} in X is called M-recursive if x_n+k = M(x_n, x_n+1,…, x_n+k?1) for all n. We prove that for a continuous strict mean value M every M-recursive sequence is convergent. We give a necessary and sufficient condition for a convergent sequence in X to be M-recursive for some continuous strict mean value M, and we characterize its limit by a functional equation. 39 B 72, 39 B 52, 40 A 05.     

Limiting -subgradient characterizations of constrained best approximation

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the set

K:=C∩{xX:-g(x)S},